Simplify Algebraic Expressions Which Expression Is Equivalent To $8 A^2 B \cdot 2 B^4 C^4$

Pleton · · 4 min read

Table Of Content

  1. Understanding the Basics
    1. The Role of Coefficients and Variables
    2. Multiplication and Exponents
  2. Breaking Down the Expression 8a2b⋅2b4c48a^2b \cdot 2b^4c^48a2b⋅2b4c4
    1. Step 1: Multiplying the Coefficients
    2. Step 2: Dealing with the Variable a
    3. Step 3: Simplifying the Variable b
    4. Step 4: Handling the Variable c
    5. Step 5: Putting It All Together
  3. The Simplified Expression: 16a2b5c416a^2b^5c^416a2b5c4
  4. Common Mistakes to Avoid
    1. Mistake 1: Incorrectly Multiplying Coefficients
    2. Mistake 2: Forgetting the Exponent Rule
    3. Mistake 3: Combining Unlike Terms
    4. Mistake 4: Overlooking Coefficients of 1
  5. Practice Problems
    1. Problem 1
    2. Problem 2
    3. Problem 3
  6. Answers to Practice Problems
    1. Answer to Problem 1
    2. Answer to Problem 2
    3. Answer to Problem 3
  7. Real-World Applications of Simplifying Expressions

Hey guys! Ever stumbled upon an algebraic expression that looks like a jumbled mess? You're not alone! Simplifying expressions is a fundamental skill in mathematics, and it's something we all grapple with at some point. Today, we're going to break down a specific problem: figuring out which expression is equivalent to 8a2b⋅2b4c48a^2b \cdot 2b^4c^4. Trust me, by the end of this article, you'll be a pro at tackling these types of problems. Let's dive in!

Understanding the Basics

Before we jump into the main problem, let's refresh some key concepts. When you see an algebraic expression, it's a combination of variables (like a, b, and c) and constants (like 8 and 2), connected by mathematical operations such as multiplication. The goal of simplifying is to rewrite the expression in its most concise form, making it easier to understand and work with. Simplifying algebraic expressions often involves combining like terms and applying the rules of exponents. It's like decluttering your room – you want everything to be neat and organized!

The Role of Coefficients and Variables

In our expression, 8a2b⋅2b4c48a^2b \cdot 2b^4c^4, we have coefficients and variables. The coefficients are the numerical parts (8 and 2), and the variables are the letters (a, b, and c). Each variable may have an exponent, which tells you how many times the variable is multiplied by itself. For example, a2a^2 means a multiplied by itself (a * a). Understanding this distinction is crucial because it guides how we manipulate the expression.

Multiplication and Exponents

When multiplying terms with the same base (like b in our expression), we use the product of powers rule: xm⋅xn=xm+nx^m \cdot x^n = x^{m+n}. This means we add the exponents. For instance, b⋅b4b \cdot b^4 becomes b1+4b^{1+4} or b5b^5. Remembering this rule is key to simplifying our expression correctly. It’s like knowing the secret handshake to the world of exponents!

Breaking Down the Expression 8a2b⋅2b4c48a^2b \cdot 2b^4c^4

Okay, let's tackle our main problem. We have 8a2b⋅2b4c48a^2b \cdot 2b^4c^4. The first step is to identify the different parts: we have coefficients (8 and 2), the variable a with an exponent, the variable b appearing twice with different exponents, and the variable c with an exponent. Our mission is to simplify this by multiplying the coefficients and combining the variables.

Step 1: Multiplying the Coefficients

The coefficients are 8 and 2. When we multiply them, we get 8⋅2=168 \cdot 2 = 16. So, our simplified expression will start with 16. This is straightforward but essential! Think of it as laying the foundation for the rest of the simplification.

Step 2: Dealing with the Variable a

The variable a appears as a2a^2. Since there are no other a terms to combine with, we simply carry a2a^2 over to our simplified expression. It's like having a unique piece that doesn't need to be combined with anything else.

Step 3: Simplifying the Variable b

Here's where the exponent rule comes into play. We have b and b4b^4. Remember, b is the same as b1b^1. Using the product of powers rule, we add the exponents: b1⋅b4=b1+4=b5b^1 \cdot b^4 = b^{1+4} = b^5. This is a crucial step, and understanding it is key to mastering algebraic simplification. It's like transforming two separate pieces into one powerful whole!

Step 4: Handling the Variable c

The variable c appears as c4c^4. Similar to a, there are no other c terms to combine with, so we carry c4c^4 over to our simplified expression. It’s another unique piece that remains unchanged.

Step 5: Putting It All Together

Now, let's combine everything we've simplified. We have the coefficient 16, the variable part a2a^2, the simplified b term b5b^5, and the variable part c4c^4. Putting these together, we get 16a2b5c416a^2b^5c^4. And that's our simplified expression! Feels good, right?

The Simplified Expression: 16a2b5c416a^2b^5c^4

So, the expression equivalent to 8a2b⋅2b4c48a^2b \cdot 2b^4c^4 is 16a2b5c416a^2b^5c^4. We got there by methodically breaking down the expression, multiplying the coefficients, and applying the product of powers rule for variables with exponents. This step-by-step approach is your best friend when tackling algebraic expressions. It's like following a recipe – each step is crucial for the final delicious result!

Common Mistakes to Avoid

Simplifying algebraic expressions can sometimes be tricky, and there are a few common mistakes you might encounter. Recognizing these pitfalls can save you a lot of headaches!

Mistake 1: Incorrectly Multiplying Coefficients

One common error is multiplying the coefficients incorrectly. For example, someone might mistakenly multiply 8 and 2 and get 10 instead of 16. Always double-check your multiplication! It's a small step that can make a big difference.

Mistake 2: Forgetting the Exponent Rule

Another frequent mistake is forgetting to add the exponents when multiplying variables with the same base. For instance, mixing up b⋅b4b \cdot b^4 and thinking it's b4b^4 instead of b5b^5. Remembering and applying the product of powers rule correctly is essential. It’s like having the right tool for the job!

Mistake 3: Combining Unlike Terms

A very common error is trying to combine terms that are not alike. For example, trying to combine a2a^2 with b5b^5. Remember, you can only combine terms that have the same variable raised to the same power. It’s like trying to fit puzzle pieces that don't match – they just won't go together!

Mistake 4: Overlooking Coefficients of 1

Sometimes, variables appear without an explicit coefficient, like b. Remember that b is the same as 1b. Overlooking this can lead to errors in simplification. Always be mindful of those invisible coefficients! They’re like the silent partners in your expression.

Practice Problems

Now that we've gone through the solution and common pitfalls, let's test your understanding with a few practice problems. The best way to master simplifying algebraic expressions is through practice, practice, practice!

Problem 1

Simplify the expression: 3x3y2⋅4xy53x^3y^2 \cdot 4xy^5

Problem 2

Which expression is equivalent to 5p2q3⋅2p4q5p^2q^3 \cdot 2p^4q?

Problem 3

Simplify: 7m4n⋅6mn37m^4n \cdot 6mn^3

Try solving these on your own. Use the step-by-step method we discussed, and pay attention to the common mistakes. The answers are below, but try to solve them without looking first!

Answers to Practice Problems

Ready to check your work? Here are the answers to the practice problems:

Answer to Problem 1

3x3y2⋅4xy5=12x4y73x^3y^2 \cdot 4xy^5 = 12x^4y^7

Answer to Problem 2

5p2q3⋅2p4q=10p6q45p^2q^3 \cdot 2p^4q = 10p^6q^4

Answer to Problem 3

7m4n⋅6mn3=42m5n47m^4n \cdot 6mn^3 = 42m^5n^4

How did you do? If you got them right, awesome! You're well on your way to mastering algebraic simplification. If you made a mistake, don't worry – that's how we learn. Go back, review the steps, and try again. Each attempt brings you closer to understanding.

Real-World Applications of Simplifying Expressions

You might be wondering,

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